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## Quantifying loss in cricket II: New measures

A few days ago I penned a wee diatribe on the use of net run rate (NRR) as a tiebreaker in cricket. Today I will outline a few alternatives and their justification. The punchline is the revised group score tables for the 2007 World Cup at the end of the post and the formulae used to compute them scattered throughout. There isn’t a single way forward that stands out, however there are plenty of simple ideas that are much more appealing than the status quo.

Recall that the essential difficulty is the failure of the NRR to incorporate loss of wickets; only overs and runs are included. Throughout the following nomenclature is used: runs scored, overs batted and wickets lost are represented by R, O and W respectively, with subscript indices A and B labelling the two teams. In this notation, the NRR (for team A) is calculated by

$\mathrm{NRR} = \frac{R_A}{O_A} - \frac{R_B}{O_B^\star}$

with $O_B^\star = O_B$ (B batting first) or $O_B = 50$ (B batting second). Here we are looking for an extension of the net run rate to a generalised measure; one aims to modify the NRR formula to include WA and WB such that a set of desiderata are fulfilled. These might be:

1. Retrievability: The measure reduces to NRR in the limit that WA = WB;
2. Sufficient reward: The winning team measures higher than the losing team;
3. Symmetry: The measure is (anti-)symmetric in A and B, so it does not matter which team has won, or who has batted first;
4. Separability: The components of the measure relating to each team are independent, so each team’s performance is evaluated without regard to the other’s; the measure is simply the comparison of these two separate evaluations.

My belief is that all of these have some intuitive appeal, i.e. they are ‘obviously’ good things to require of such a measure. However, they also seem to be mutually incompatible, so this intuition is wrong at some level. Different selections of which to satisfy lead to different measures. Here is a simple example:

$M_1 = \frac{R_A}{(1+W_A)O_A} - \frac{R_B}{(1+W_B)O_B}$;

perhaps this is the simplest sensible modification. Losing wickets should decrease the measure, so divide by wickets as well as overs; because wickets can be zero, 1 + W must be used. Note this doesn’t reduce to NRR when wickets lost are level, nor does it guarantee that the winner of the match performs better (the loser could have the ‘moral victory’). However it is well-behaved in every other sense. Now consider

$M_2 = \frac{1+W_B}{1+W_A}\left(\frac{R_A}{O_A} - \frac{R_B}{O_B}\right)$.

This comes closer to NRR when wickets are level and guarantees that the winner (team A in this case) has a positive measure, but it isn’t symmetric (so B‘s measure has to defined to be -1 times this number, as in the case of NRR) or separable. I’ve no doubt you can think of several more possibilities without trouble; I’ve decided not to produce a laundry list. As far as I’m concerned, the holy grail would be a measure simultaneously satisfying conditions (2) – (4) above, but I haven’t found it yet. For the time being, the litmus test of these new measures is to see if they rank teams in a way that feels better than NRR. So without further ado, the World Cup games played thus far:

 # A B RA OA WA RB OB WB NRR M1 M2 1 WI Pak 241 50 9 187 47.33 10 1.08 0.12 0.96 2 Aus Sco 334 50 6 131 40.17 10 4.06 0.66 5.37 3 Can Ken 199 50 10 203 43.33 3 -0.70 -0.81 -1.94 4 SL Ber 321 50 6 203 24.67 10 4.86 0.63 5.12 5 Ire Zim 221 50 9 221 50 10 0 0.04 0 6 Eng NZ 209 50 7 210 41 4 -0.94 -0.5 -1.51 7 SA Ned 353 40 3 132 40 9 5.53 1.88 13.81 8 Ind Ban 191 49.5 10 192 48.5 5 -0.10 -0.31 -0.18 9 Pak Ire 132 45.67 10 133 48.5 5 -0.30 -0.14 -0.41 10 Aus Ned 358 50 5 129 26.83 10 4.58 0.76 4.31

The numbers in the last three columns are the NRR for the first-listed team; the opposition NRR is -1 times these values. Thus, it is possible to rank teams based on wins, draws and losses first, and by one of these measures second. Thus, the four groups are:

 Team Points NRR M1 M2 Team Points NRR M1 M2 Australia 4 8.64 1.42 9.68 Sri Lanka 2 4.86 0.63 5.12 South Africa 2 5.53 1.88 13.81 Bangladesh 2 0.1 0.31 0.18 Scotland 0 -4.06 -0.66 -5.37 India 0 -0.1 -0.31 -0.18 Netherlands 0 -10.11 -2.64 -18.12 Bermuda 0 -4.86 -0.63 -5.12
 Team Points NRR M1 M2 Team Points NRR M1 M2 New Zealand 2 0.94 0.5 1.51 Ireland 3 0.3 0.18 0.41 Kenya 2 0.7 0.81 1.94 West Indies 2 1.08 0.12 0.96 Canada 0 -0.7 -0.81 -1.94 Zimbabwe 1 0 -0.04 0 England 0 -0.94 -0.5 -1.51 Pakistan 0 -1.38 -0.26 -1.37

Group C (bottom-left) shows immediately how the inclusion of wickets changes how the teams are ranked – this after only the first set of games. I’m optimistic that further deviation will be visible as the preliminary round progresses, so this table will be updated. The choices of measure made above are arbitrary to a degree, but the inclusion of wickets in the ranking of a team’s performance clearly makes them fairer for ranking. Teams whose performance has been misjudged by the NRR, and who are lower on the table than they should be as a result, have every reason to be upset.